A noncommutative version of Lie algebras: Leibniz algebras. (Une version non commutative des algèbres de Lie: les algèbres de Leibniz.) (French) Zbl 0806.55009

Loday, Jean-Louis

Enseign. Math., II. Sér. 39, No. 3-4, 269-293 (1993).

A Leibniz algebra over a commutative ring \(\mathbb{K}\) is a \(\mathbb{K}\)-module \(g\) together with a bilinear map \([-,-]: g\times g\to g\) satisfying for all \(x\), \(y\), \(z\) in \(G\), \([x, [y,z]] - [[x, y],z]+ [[x, z],y]=0\). The paper introduces this new algebraic structure and presents some interesting properties and connections with other theories. The main motivation is the existence of a homology theory \(HL_*\) that gives in particular new algebraic invariants for Lie algebras.
This paper is clearly written and gives a lot of new and interesting questions in homological algebra.

Reviewer: Yves Félix (Louvain-La-Neuve)

MSC:

17A32	Leibniz algebras
17B55	Homological methods in Lie (super)algebras
18G60	Other (co)homology theories (MSC2010)
19D55	\(K\)-theory and homology; cyclic homology and cohomology

Keywords:

Hochschild homology; Leibniz algebra; algebraic invariants for Lie algebras